Question

find the general solution..
1. y''-2y'-2y=0
2. y''''-8y''-9y=0
3. y''''+13y''+36y=0
4. y'''''-15y''''+84y'''-220y''-275y'=0
5. y'''''+12y''''+104y'''+408y''-1156y'=0

Answer 1

man I wish my prof taught me this... I do know that there's a second order differential equation... OH WAIT a sec it could be one of those replace the y'' with r... so it would be r^2-2r-2 = 0. you use the discriminant to see if it factors... there's three types of equations... one with a single root, one with two real roots, and one with a complex root. should there be a complex root you need to use the equation that involves cosx and sinx.

Answer 2

complex root problems require the usage of the quadratic formula

Answer 3

These are pretty easy if you see that e^(ax) has a first derivative of a*e^(ax). Just plug these all in and divide out e^(ax) leaving the a's behind to solve for the a in your function, y=e^(ax).

I'll do #2 to show you:

a^4(e^ax)-8a^2(e^ax)-9(e^ax)=0

divide both sides by e^(ax)

a^4-8a^2-9=0

This looks tough, but really you can think of a^4 as being (a^2)^2 and it looks like a normal quadratic:

(a^2)^2-8(a^2)-9=0

Now you can factor:

(a^2-9)(a^2+1)=0

Solve for a, and you will find that you have multiple answers, some of which might even be complex numbers. Depending on your class, you can exclude them unless they want them, not really that crazy.

But the interesting thing is that now you can add up all your answers and it should still be right.

y=e^3x
y=e^-3x

test it out yourself to make sure.